2B: 2nd Order Spatial Point Patterns Analysis

In this exercise, we will learn to apply 2nd-order spatial point pattern analysis methods in R, including G, F, K, and L functions, to evaluate spatial point distributions and perform hypothesis testing using the spatstat package.

Author

Teng Kok Wai (Walter)

Published

August 29, 2024

Modified

September 5, 2024

1 Exercise 2B Reference

R for Geospatial Data Science and Analytics - 5  2nd Order Spatial Point Patterns Analysis Methods

2 Overview

Spatial Point Pattern Analysis is the evaluation of the pattern or distribution, of a set of points on a surface. The point can be location of:

  • events such as crime, traffic accident and disease onset, or
  • business services (coffee and fastfood outlets) or facilities such as childcare and eldercare.

Using appropriate functions of spatstat, this hands-on exercise aims to discover the spatial point processes of childecare centres in Singapore.

The specific questions we would like to answer are as follows:

  • are the childcare centres in Singapore randomly distributed throughout the country?
  • if the answer is not, then the next logical question is where are the locations with higher concentration of childcare centres?

This hands-on exercise continues from Hands-on Exercise 2A

Here’s the improved section with relevant information presented in table format:

3 The Data

The following data sources will be used in this exercise:

Dataset Description Source Format
CHILDCARE Point data containing location and attributes of childcare centres. Data.gov.sg GeoJSON
MP14_SUBZONE_WEB_PL Polygon data representing URA 2014 Master Plan Planning Subzone boundaries. Data.gov.sg ESRI Shapefile
CoastalOutline Polygon data showing the national boundary of Singapore. Singapore Land Authority (SLA) ESRI Shapefile

4 Installing and Loading the R Packages

The following R packages will be used in this exercise:

Package Purpose Use Case in Exercise
sf Importing, managing, and processing vector-based geospatial data. Handling vector geospatial data in R.
spatstat Tools for point pattern analysis. Performing 1st- and 2nd-order spatial point pattern analysis and deriving kernel density estimation (KDE).
raster Reading, writing, manipulating, analyzing, and modeling of gridded spatial data (raster). Converting image outputs generated by spatstat into raster format.
maptools Tools for manipulating geographic data. Converting spatial objects into ppp format for use with spatstat.
tmap Functions for plotting cartographic quality static or interactive maps. Creating static and interactive point pattern maps.

To install and load these packages in R, use the following code:

pacman::p_load(sf, raster, spatstat, tmap, tidyverse)

5 Spatial Data Wrangling

5.1 Importing the Spatial Data

To import the three geographical datasets, we will use st_read() from sf.

childcare_sf <- st_read("data/child-care-services-geojson.geojson")
Reading layer `child-care-services-geojson' from data source 
  `/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex02/data/child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
sg_sf <- st_read(dsn = "data", layer="CostalOutline")
Reading layer `CostalOutline' from data source 
  `/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex02/data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
mpsz_sf <- st_read(dsn = "data",
                layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex02/data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

5.2 Inspect and Reproject to Same Projection System

Before we can use these data for analysis, it is important for us to ensure that they are projected in same projection system.

First, we check the childcare dataset.

st_crs(childcare_sf)
Coordinate Reference System:
  User input: WGS 84 
  wkt:
GEOGCRS["WGS 84",
    DATUM["World Geodetic System 1984",
        ELLIPSOID["WGS 84",6378137,298.257223563,
            LENGTHUNIT["metre",1]]],
    PRIMEM["Greenwich",0,
        ANGLEUNIT["degree",0.0174532925199433]],
    CS[ellipsoidal,2],
        AXIS["geodetic latitude (Lat)",north,
            ORDER[1],
            ANGLEUNIT["degree",0.0174532925199433]],
        AXIS["geodetic longitude (Lon)",east,
            ORDER[2],
            ANGLEUNIT["degree",0.0174532925199433]],
    ID["EPSG",4326]]

This dataset is using the WGS84 crs. We will reproject all the dataset to SVY21 crs for standardization and analysis.

childcare_sf <- st_transform(childcare_sf , crs = 3414)
st_crs(childcare_sf)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

The childcare dataset has been reprojected to SVY21 successfully.

Next, we inspect the Coastal Outline dataset.

st_crs(sg_sf)
Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

Notice that this dataset is using SVY21 crs, however the ID provided is EPSG:9001 does not match the intended ID, EPSG:3414 of SVY21. In this case, we will set the crs to the correct ID using the code block below.

sg_sf <- st_set_crs(sg_sf, 3414)
st_crs(sg_sf)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Similarly, we will inspect the Master Plan Subzone Dataset.

st_crs(mpsz_sf)
Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

Since the ID is also EPSG:9001, we will set the crs to EPSG:3414 too.

mpsz_sf <- st_set_crs(mpsz_sf, 3414)
st_crs(mpsz_sf)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

6 Mapping the Geospatial Datasets

After checking the referencing system of each geospatial data data frame, it is also useful for us to plot a map to show their spatial patterns.

6.1 Static Map

# add polygon layer of the coastal outline of sg island
tm_shape(sg_sf)+ tm_polygons() +
# add polygon layer of the subzone based on sg masterplan
tm_shape(mpsz_sf) + tm_polygons() +
# add dot layer to show the locations of childcare centres
tm_shape(childcare_sf) + tm_dots() +
tm_layout()

When all the 3 datasets are overlayed together, it shows the locations of childcare centres on the Singapore island. Since all the geospatial layers are within the same map context, it means their referencing system and coordinate values are referred to similar spatial context. This consistency is crucial for accurate geospatial analysis.

6.2 Interactive Map

Alternatively, we can also prepare a pin map by using the code block below.

tmap_mode('view')

# tm_basemap("Esri.WorldGrayCanvas") +
# tm_basemap("OpenStreetMap") +
tm_basemap("Esri.WorldTopoMap") +
tm_shape(childcare_sf) +
  tm_dots(alpha = 0.5)
tmap_mode('plot')

In interactive mode, tmap uses the Leaflet for R API, allowing you to freely navigate, zoom, and click on features for detailed information. You can also change the map’s background using layers like ESRI.WorldGrayCanvas, OpenStreetMap, and ESRI.WorldTopoMap, with ESRI.WorldGrayCanvas as the default.

Tip

Remember to switch back to plot mode after interacting to avoid connection issues and limit interactive maps to fewer than 10 in RMarkdown documents for Netlify publishing.

7 Geospatial Data Wrangling

While simple feature data frames are becoming more popular compared to sp’s Spatial* classes, many geospatial analysis packages still require data in the Spatial* format. This section will show you how to convert a simple feature data frame to sp’s Spatial* class.

7.1 Converting sf data frames to sp’s Spatial* class

The code block below uses as_Spatial() of sf package to convert the three geospatial data from simple feature data frame to sp’s Spatial* class.

childcare <- as_Spatial(childcare_sf)
mpsz <- as_Spatial(mpsz_sf)
sg <- as_Spatial(sg_sf)

After the sf dataframe to sp Spatial* conversion, let’s inspect the Spatial* classes.

childcare
class       : SpatialPointsDataFrame 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 2
names       :    Name,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Description 
min values  :   kml_1, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>018989</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>1, MARINA BOULEVARD, #B1 - 01, ONE MARINA BOULEVARD, SINGAPORE 018989</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>THE LITTLE SKOOL-HOUSE INTERNATIONAL PTE. LTD.</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>08F73931F4A691F4</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
max values  : kml_999,                  <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>829646</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>200, PONGGOL SEVENTEENTH AVENUE, SINGAPORE 829646</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td>Child Care Services</td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>RAFFLES KIDZ @ PUNGGOL PTE LTD</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>379D017BF244B0FA</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
mpsz
class       : SpatialPolygonsDataFrame 
features    : 323 
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 15
names       : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C,          INC_CRC, FMEL_UPD_D,     X_ADDR,     Y_ADDR,    SHAPE_Leng,    SHAPE_Area 
min values  :        1,          1, ADMIRALTY,    AMSZ01,      N, ANG MO KIO,         AM, CENTRAL REGION,       CR, 00F5E30B5C9B7AD8,      16409,  5092.8949,  19579.069, 871.554887798, 39437.9352703 
max values  :      323,         17,    YUNNAN,    YSSZ09,      Y,     YISHUN,         YS,    WEST REGION,       WR, FFCCF172717C2EAF,      16409, 50424.7923, 49552.7904, 68083.9364708,  69748298.792 
sg
class       : SpatialPolygonsDataFrame 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 4
names       : GDO_GID, MSLINK, MAPID,              COSTAL_NAM 
min values  :       1,      1,     0,             ISLAND LINK 
max values  :      60,     67,     0, SINGAPORE - MAIN ISLAND 

The geospatial data have been converted into their respective sp’s Spatial* classes.

7.2 Converting the Generic sp Format into spatstat’s ppp Format

Now, we will use as.ppp() function of spatstat to convert the spatial data into spatstat’s ppp object format.

childcare_ppp <- as.ppp(childcare_sf)
childcare_ppp
Marked planar point pattern: 1545 points
marks are of storage type  'character'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units

Let’s examine the difference by plotting childcare_ppp:

plot(childcare_ppp)

We can also view the summary statistics of the newly created ppp object by using the code block below.

summary(childcare_ppp)
Marked planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1545 character character 

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units
Tip

Be aware of the warning message regarding duplicates. In spatial point pattern analysis, duplicates can be a significant issue. The statistical methods used for analyzing spatial point patterns often assume that the points are distinct and non-coincident.

7.3 Handling Duplicated Points

We can check the duplication in a ppp object by using the duplicated function with different configurations.

Tip

Tips on using duplicated

If rule=“spatstat” (the default), two points are deemed identical if their coordinates are equal according to ==, and their marks are equal according to ==. This is the most stringent possible test. If rule=“unmark”, duplicated points are determined by testing equality of their coordinates only, using ==. If rule=“deldir”, duplicated points are determined by testing equality of their coordinates only, using the function duplicatedxy in the package deldir, which currently uses duplicated.data.frame. Setting rule=“deldir” will ensure consistency with functions in the deldir package.

see R: Determine Duplicated Points in a Spatial Point Pattern

# duplicated(childcare_ppp)
# any(duplicated(childcare_ppp))
rules <- c("spatstat", "deldir", "unmark")

duplicate_counts <- list()
for (rule in rules) {
  duplicates <- duplicated(childcare_ppp, rule = rule)
  num_duplicates <- sum(duplicates)
  duplicate_counts[[rule]] <- num_duplicates
}

print(duplicate_counts)
$spatstat
[1] 0

$deldir
[1] 0

$unmark
[1] 74

Using unmark, we can find 74 duplicates.

We can also use anyDuplicated(x) as it is a faster version of any(duplicated(x)).

sum(anyDuplicated(childcare_ppp, rule="unmark"))
[1] 0

To count the number of coincident points, we will use the multiplicity() function as shown in the code block below. see R: Multiplicity for more info.

multiplicity(childcare_ppp)

If we want to know how many locations have more than one point event:

sum(multiplicity(childcare_ppp) > 1)
[1] 0
# double check
coincident_points <- duplicated(childcare_ppp,  rule="deldir")
coincident_coordinates <- childcare_ppp[coincident_points]
print(coincident_coordinates)
Marked planar point pattern: 0 points
marks are of storage type  'character'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units

The output shows that there are 128(???) 0 duplicated point events.

tmap_mode('view')
tm_basemap("Esri.WorldTopoMap") +
tm_shape(childcare) +
  tm_dots(alpha=0.4,
          size=0.05)
tmap_mode('plot')

7.4 How to Spot Duplicate Points on the Map

There are three ways to overcome this problem. The easiest way is to delete the duplicates. But, that will also mean that some useful point events will be lost.

The second solution is use jittering, which will add a small perturbation to the duplicate points so that they do not occupy the exact same space.

The third solution is to make each point “unique” and then attach the duplicates of the points to the patterns as marks, as attributes of the points. Then you would need analytical techniques that take into account these marks.

7.4.1 Jittering

The code block below implements the jittering approach.

childcare_ppp_jit <- rjitter(childcare_ppp,
                             retry=TRUE,
                             nsim=1,
                             drop=TRUE)
any(duplicated(childcare_ppp_jit))
[1] FALSE

7.5 Creating owin Object

When analysing spatial point patterns, it is a good practice to confine the analysis with a geographical area like Singapore boundary. In spatstat, an object called owin is specially designed to represent this polygonal region.

The code block below is used to covert sg SpatialPolygon object into owin object of spatstat.

sg_owin <- as.owin(sg_sf)

The output object can be displayed by using plot() function:

plot(sg_owin)

And using summary() function of Base R:

summary(sg_owin)
Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

7.6 Combining Point Events Object and Owin Object

For the last step of geospatial data wrangling, we will extract childcare events that are located within Singapore by using the code block below.

childcareSG_ppp = childcare_ppp[sg_owin]

The output object combined both the point and polygon feature in one ppp object class as shown below.

summary(childcareSG_ppp)
Marked planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1545 character character 

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401
plot(childcareSG_ppp)

7.6.1 Extracting Study Area

The code block below will be used to extract the target planning areas.

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")

Plotting the target planning areas:

par(mfrow=c(2,2))
plot(pg, main = "Punggol")

plot(tm, main = "Tampines")

plot(ck, main = "Choa Chu Kang")

plot(jw, main = "Jurong West")

7.6.2 Converting sf objects into owin objects

Now, we will convert these sf objects into owin objects that is required by spatstat.

pg_owin = as.owin(pg)

tm_owin = as.owin(tm)

ck_owin = as.owin(ck)

jw_owin = as.owin(jw)

7.6.3 Combining Childcare Points and the Study Area

To extract childcare that is within the specific region for analysis, we can use:

childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]

Next, rescale() function is used to transform the unit of measurement from meter to kilometer.

childcare_pg_ppp.km = rescale(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale(childcare_jw_ppp, 1000, "km")

The code block below is used to plot the four study areas and the locations of the childcare centres.

par(mfrow=c(2,2))

plot(childcare_pg_ppp.km, main="Punggol")
plot(childcare_tm_ppp.km, main="Tampines")
plot(childcare_ck_ppp.km, main="Choa Chu Kang")
plot(childcare_jw_ppp.km, main="Jurong West")

8 Second-Order Spatial Point Pattern Analysis

In this section, we will analyze spatial point patterns using various functions: G-Function, F-Function, K-Function, and L-Function.

8.1 Analysing Spatial Point Process Using G-Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, we will learn how to compute G-function estimation by using Gest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.

8.1.1 Choa Chu Kang Planning Area

8.1.1.1 Computing G-function Estimation

To compute G-function using Gest() of spatstat package:

Note

correction is an optional argument in Gest()

Optional. The edge correction(s) to be used to estimate . A vector of character strings selected from “none”, “rs”, “km”, “Hanisch” and “best”. Alternatively correction=“all” selects all options.

see Gest function - RDocumentation

# rs and border has the same effect
G_CK = Gest(childcare_ck_ppp, correction = "rs")
G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

We can also use the “all” option in correction to display all forms of edge corrections from “none”, “rs”, “km”, “Hanisch” and “best”.

G_CK_all = Gest(childcare_ck_ppp, correction = "all")
plot(G_CK_all, xlim=c(0,500))

8.1.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.

\(H_1\)= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with G-function:

Note

The envelope function calculates overall and pointwise confidence envelopes for a curve based on bootstrap replicates of the curve evaluated at a number of fixed points.

In other words, It helps you determine if your observed spatial pattern (e.g., locations of points in a study area) is significantly different from what you would expect under a random or theoretical distribution.

How It Works: 1. Simulate Data: It generates multiple simulated datasets (often by randomizing the locations of points) based on the null hypothesis (e.g., complete spatial randomness). 2. Compute Statistics: For each simulated dataset, it computes a spatial statistic (e.g., G-function, F-function) and creates a distribution of these statistics. 3. Compare: It compares the observed statistic from your actual data to the distribution of statistics from the simulated datasets. 4. Envelope Plot: It plots the range (envelope) of the simulated statistics along with the observed statistic, allowing you to see if your observed statistic falls outside the range of what is expected under the null hypothesis.

When to Use It? Use the envelope function when you want to:

  1. Test if the observed spatial pattern deviates significantly from a random pattern or other theoretical patterns.
  2. Assess the statistical significance of spatial features or clustering in your data.
G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_CK.csr)

8.1.2 Tampines Planning Area

8.1.2.1 Computing G-function Estimation

We will use the best edge correction for this example.

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

G_tm_all = Gest(childcare_tm_ppp, correction = "all")
plot(G_tm_all)

8.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.

\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

The code block below is used to perform the hypothesis testing.

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_tm.csr)

8.3 Analysing Spatial Point Process Using F-Function

The F function estimates the empty space function F(r) or its hazard rate h(r) from a point pattern in a window of arbitrary shape. In this section, we will learn how to compute F-function estimation by using Fest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.

8.3.1 Choa Chu Kang Planning Area

8.3.1.1 Computing F-function estimation
Note

Fest() has the same correction option as Gest().

F_CK_all = Fest(childcare_ck_ppp, correction = "all")
plot(F_CK_all)

8.3.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.

\(H_1\) = The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with F-function:

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_CK.csr)

8.3.2 Tampines Planning Area

8.3.2.1 Computing F-function estimation

Monte Carlo test with F-function:

F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)

8.3.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.

\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

The code block below is used to perform the hypothesis testing.

F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_tm.csr)

8.4 Analysing Spatial Point Process Using K-Function

K-function measures the number of events found up to a given distance of any particular event. In this section, you will learn how to compute K-function estimates by using Kest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.

Note

Kest()’s correction option is different fromGest()andFest()`.

correction: Optional. A character vector containing any selection of the options “none”, “border”, “bord.modif”, “isotropic”, “Ripley”, “translate”, “translation”, “rigid”, “none”, “good” or “best”. It specifies the edge correction(s) to be applied. Alternatively correction=“all” selects all options.

see Kest function - RDocumentation

8.4.1 Choa Chu Kang Planning Area

8.4.1.1 Computing K-Function Estimate
K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

8.4.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.

\(H_1\) = The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code block below is used to perform the hypothesis testing.

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

8.4.2 Tampines Planning Area

8.4.2.1 Computing K-function Estimation
K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r,
     ylab= "K(d)-r", xlab = "d(m)",
     xlim=c(0,1000))

8.5 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.

\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code block below is used to perform the hypothesis testing.

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_tm.csr, . - r ~ r,
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

8.6 Analysing Spatial Point Process Using L-Function

In this section, you will learn how to compute L-function estimation by using Lest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.

8.6.1 Choa Chu Kang Planning Area

8.6.1.1 Computing L-function Estimation
Note

Lest() has the same correction options as Kest().

L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r,
     ylab= "L(d)-r", xlab = "d(m)")

8.6.1.2 Performing Complete Spatial Randomness Test

\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.

\(H_1\) = The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value if smaller than alpha value of 0.001.

The code block below is used to perform the hypothesis testing.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

8.6.2 Tampines Planning Area

8.6.2.1 Computing L-function Estimate
L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r,
     ylab= "L(d)-r", xlab = "d(m)",
     xlim=c(0,1000))

8.7 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.

\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below will be used to perform the hypothesis testing

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_tm.csr, . - r ~ r,
     xlab="d", ylab="L(d)-r", xlim=c(0,500))